不定式的運演算法

A. 高一不等式運演算法則

不等式兩邊相加或相減同一個數或式子，不等號的方向不變。（移項要變號）

不等式兩邊相乘或相除同一個正數，不等號的方向不變。（相當系數化1，這是得正數才能使用）

不等式兩邊乘或除以同一個負數，不等號的方向改變。（÷或×1個負數的時候要變號）

確定解集

1.比兩個值都大，就比大的還大(同大取大）；

2.比兩個值都小，就比小的還小(同小取小）；

3.比大的大，比小的小，無解（大大小小取不了）；

4.比小的大，比大的小，有解在中間（小大大小取中間）。

三個或三個以上不等式組成的不等式組，可以類推。

不等式的特殊性質

不等式性質1：不等式的兩邊同時加上（或減去）同一個數（或式子），不等號的方向不變；

不等式性質2：不等式的兩邊同時乘（或除以）同一個正數，不等號的方向不變；

不等式性質3：不等式的兩邊同時乘（或除以）同一個負數，不等號的方向變。 總結：當兩個正數的積為定值時，它們的和有最小值；當兩個正數的和為定值時，它們的積有最大值。

B. 不定式計算中錯位微分法是什麼

有理函數部分分式分解待定系數法+留數法。

不定積分結果 結果不唯一求導驗證應該能夠提高湊微分的計算能力先寫別問唉。

舉報數字帝國GG泛濫但是是一個計算器網頁。

。。

C. 動詞不定式

後加動詞不定式的動詞有：
afford aim appear agree arrange ask be decide bother care choose come dare demand desire determine expect elect endeavor hope fail happen help hesitatelearn long mean manage offer ought plan prepare pretend promise refuse seem tend wait wish undertake
舉例：
The driver failed to see the other car in time.
司機沒能及時看見另一輛車。
I happen to know the answer to your question.
我碰巧知道你那道問題的答案。

後加動名詞（doing）的動詞有：
admit appreciate avoid complete consider delay deny detest enre enjoy escape prevent fancy finish imagine mind miss postpone practise recall resent resist resume risk suggest face include stand understand forgive keep
舉例：
(1) Would you mind turning down your radio a little, please?
(2) The squirrel was lucky that it just missed being caught.

片語後接doing
admit to prefer…to be used to lead to devote oneself to object to stick to be busy look forward to (to為介詞） no goodn no use It's worth… as well as
can't help It's no use /good be tired of be fond of be capable of be afraid of be proud of think of / about hold off put offkeep on insist on count on / upon
set about be successful in good at
take up give up burst out prevent … from…
Find 的特殊用法：
Find 後可用分詞做賓補，或先加形式賓語，再加形容詞，最後加帶to 的動詞不定式。find後也可帶一個從句。此類動詞還有get，have。
I found him lying on the ground.
I found it important to learn.
I found that to learn English is important.
典型例題:
The next morning she found the man ___ in bed，dead.
A. lying B. lie C. lay D. laying
答案：A.find的賓語後面，用分詞或分詞短語，起賓語補足語作用。現在分詞表達主動，也表達正在進行，過去分詞表達被動。

省to的動詞不定式（即動詞原形）
1） 情態動詞 ( 除ought 外，ought to)：
2） 使役動詞 let, have, make：
3） 感官動詞 see, watch, look at, notice , observe, hear, listen to, smell, feel, find 等後作賓補，省略to。
注意：在被動語態中則to 不能省掉。
I saw him dance.
=He was seen to dance.
The boss made them work the whole night.
=They were made to work the whole night.
4） would rather，had better：
5） Why… / why not…：
6） help 可帶to，也可不帶to, help sb (to) do sth：
7） but和except：but前是動詞do時，後面出現的動詞用不帶to的動詞不定式。
8） 由and, or和than連接的兩個不定式，第二個to 可以省去：
9） 通常在discover, imagine, suppose, think, understand等詞後，可以省去to be：
He is supposed (to be) nice. 他應該是個好人。
舉例：
He wants to move to France and marry the girl.
He wants to do nothing but go out.
典型例題
1) ---- I usually go there by train.
---- Why not ___ by boat for a change?
A. to try going B. trying to go C. to try and go D. try going
答案：D. why not 後面接不帶to 的不定式，因此選D。
2） Paul doesn't have to be made ___. He always works hard.
A. learn B. to learn C. learned D. learning
答案：B. make後接不帶to 的動詞不定式，當其用於被動時，to 不可省略。

動名詞與不定式的關系
1） 動名詞與不定式的區別：
動名詞表達的是： 狀態，性質，心境，抽象，經常性，已發生的
不定式表達的是： 目的，結果，原因，具體，一次性，將發生的
2） 接不定式或動名詞，意義相同。
3） 動名詞與不定式語義不同的有11 組：
1 stop to do stop doing
stop to do 停止，中斷做某事後去做另一件事。
stop doing 停止做某事。
They stop to smoke a cigarette. 他們停下來，抽了根煙。
I must stop smoking. 我必須戒煙了。
典型例題
She reached the top of the hill and stopped ___ on a big rock by the side of the path.
A. to have rested B. resting C. to rest D. rest
答案：C。由題意可知，她到了山頂，停下來在一個路邊的大石頭上休息。因此，應選擇"stop to do sth. 停下來去做另一件事"。而不僅僅是爬山動作的終止，所以stop doing sth.不正確。

2 forget to do forget doing
forget to do 忘記要去做某事。 (未做)
forget doing 忘記做過某事。 (已做)
The light in the office is stil on. He forgot to turn it off.
辦公室的燈還在亮著，它忘記關了。(沒有做關燈的動作)
He forgot turning the light off.
他忘記他已經關了燈了。 ( 已做過關燈的動作)
Don't forget to come tomorrow.
別忘了明天來。 (to come動作未做)
典型例題
---- The light in the office is still on.
---- Oh，I forgot___.
A. turning it off B. turn it off
C. to turn it off D. having turned it off
答案：C。由the light is still on 可知燈亮著，即關燈的動作沒有發生，因此用forget to do sth.
而forget doing sth表示燈已經關上了，而自己忘記了這一事實。此處不符合題意。

3 remember to do remember doing
remember to do 記得去做某事 (未做)
remember doing 記得做過某事 (已做)
Remember to go to the post office after school.
記著放學後去趟郵局。
Don't you remember seeing the man before?
你不記得以前見過那個人嗎?

4 regret to do regret doing
regret to do 對要做的事遺憾。 (未做)
regret doing 對做過的事遺憾、後悔。 (已做)
I regret to have to do this, but I have no choice.
我很遺憾必須這樣去做，我實在沒有辦法。
I don't regret telling her what I thought.
我不為告訴她我的想法而後悔。
典型例題
---You were brave enough to raise objections at the meeting.
---Well, now I regret ___ that.
A. to do B. to be doing C. to have done D. having done
答案：D。regret having done sth. 對已發生的事感到遺憾。regret to do sth. 對將要做的事感到遺憾。本題為對已說的話感到後悔，因此選D。

5 cease to do cease doing
cease to do 長時間，甚至永遠停做某事。
cease doing 短時停止做某事，以後還會接著做。
That department has ceased to exist forever.
那個部門已不復存在。
The girls ceased chatting for a moment when their teacher passed by.
姑娘們在老師走過時，停了會聊天。

6 try to do try doing
try to do 努力，企圖做某事。
try doing 試驗，試著做某事。
You must try to be more careful.
你可要多加小心。
I tried gardening but didn't succeed.
我試著種果木花卉，但未成功。

7 go on to do go on doing
go on to do 做了一件事後，接著做另一件事。
go on doing 繼續做原來做的事。
After he had finished his maths，he went on to do his physics.
做完數學後，他接著去做物理。
Go on doing the other exercise after you have finished this one.
作完這個練習後，接著做其他的練習。

8 be afraid to do be afraid of doing
be afraid to do 不敢，膽怯去做某事，是主觀上的原因不去做，意為"怕"；
be afraid of doing 擔心出現doing的狀況、結果。 doing 是客觀上造成的，意為"生怕，恐怕"。
She was afraid to step further in grass because she was afraid of being bitten by a snake.
她生怕被蛇咬著，而不敢在草叢中再走一步。
She was afraid to wake her husband.
她不敢去叫醒她丈夫。
She was afraid of waking her husband.
她生怕吵醒她丈夫。

9 be interested to do be interested in doing
interested to do 對做某事感興趣，想了解某事。
interested in doing 對某種想法感興趣，doing 通常為想法。
I shall be interested to know what happens.
我很想知道發生了什麼事。 (想了解)
I'm interested in working in Switzerland. Do you have any idea about that?
我對在瑞士工作感興趣。你想過這事嗎? (一種想法)

10 mean to do mean doing
mean to do 打算、想
mean doing 意味著
I mean to go, but my father would not allow me to.
我想去，但是我父親不肯讓我去。
To raise wage means increasing purchasing power.
贈加工資意味著增加購買力。

11 begin/ start to do begin/ start doing
begin / start to do sth
begin / start doing sth.

12 感官動詞+doing/to do
感官動詞 see, watch, observe, notice, look at, hear, listen to, smell, taste, feel + do 表示動作的完整性，真實性；+doing 表示動作的連續性，進行性
I saw him work in the garden yesterday.
昨天我看見他在花園里幹活了。(強調"我看見了"這個事實)
I saw him working in the garden yesterday.（強調"我見他正幹活"這個動作）
昨天我見他正在花園里幹活。
典型例題
1）They knew her very well. They had seen her ___ up from childhood.
A. grow B. grew C. was growing D. to grow
答案：A。因題意為，他們看著她長大，因此強調的是成長的過程，而非正在長的動作，因此用see sb do sth 的句型。
2）The missing boy was last seen ___ near the river.
A. playing B. to be playing C. play D. to play
答案A. 本題強調其動作，正在河邊玩，應此用see sb. doing sth句型。

D. 什麼是動詞不定式

動詞不定式的簡介
動詞不定式是一種非限定性動詞，由to＋動詞原形構成，但在有些情況下to可以省略。它還是屬於動詞，本身可以帶賓語和狀語。動詞不定式在句中可以作主語、賓語、表語、定語和狀語，還可用在復合結構中，而且有完成式、進行式、完成進行式和否定形式。動詞不定式的被動形式除了一般形式外還有其完成式和進行式。

動詞不定式的時態、語態
動詞不定式可以作以上各種成分，但它畢竟是動詞，所以有動詞的屬性
動詞不定式及其短語還可以有自己的賓語、狀語，雖然動詞不定式在語法上沒有表面上的直接主語，但它表達的意義是動作，這一動作一定由使動者發出。這一使動者我們稱之為 邏輯主語，其形式如下：
主動形式 被動形式
一般式 （not） to make （not） to be made
完成式 （not） to have made （not） to have been made
進行式 （not） to be making
完成進行式 （not） to have been making
(1)語態
如果動詞不定式的邏輯主語是這個不定式所表示的動作的承受者，不定式一般要用被動語態形式。如：
It's a great honour to be invited to Mary's birthday party.（不定式作主語）
It was impossible for lost time to be made up.（不定式作主語）
I wish to be sent to work in the country.（不定式作賓語）
Can you tell me which is the car to be repaired?（不定式作定語）
He went to the hospital to be examined.（不定式作狀語）
在There be結構中，修飾主語的不定式可用被動，也可用主動。如：There are still many things to t ake care of (to be taken care of).但有時兩種形式表達的意思不同，如：These is nothing to do now.( We have nothing to do now.) There is nothing to be done now.(We can do nothing now.)

(2)時態
1) 現在時：一般現在時表示的動詞，有時與謂語動詞表示的動作同時發生，有時發生在謂語動詞的動作之後。
He seems to know this.
I hope to see you again. = I hope that I'll see you again. 我希望再見到你。
2) 完成時：表示的動作發生在謂語動詞表示的動作之前。
I'm sorry to have given you so much trouble.
He seems to have caught a cold.
3) 進行時： 表示動作正在進行，與謂語動詞表示的動作同時發生。
He seems to be eating something.
4) 完成進行時：
She is known to have been wreaking on the problem for many years.

疑問詞＋不定式結構
疑問詞who,what,which,when,where,whether,how後可接不定式構成不定式短語，在句中作主語、賓語、表 語等。如：
①When to leave for London has not been decided yet. （不定式在句子中做主語）
②Mr. Smith didn't know whether to leave or stay there. （不定式在句子中做賓語）
③I asked Professor Xu how to learn English well. （不定式在句子中做間接賓語）
④The question was where to get the medicine needed. （不定式在句子中表語）
以上例句中疑問詞＋不定式部分，均可轉換為相應的從句形式。如：①When we shall leave…③…how I
could learn…
經常在這種結構中使用的動詞有：consider,decide,discover,explain,find out,forget,hear,know,lea rn,observe,understand,wonder等。

動詞不定式的語法功能
一、作賓語
1) 動詞+ 不定式
afford. aim. appear. agree. arrange . ask . be . decide. bother . care . choose . come. dare. demand. desire. determine . expect. elect . endeavor .hope. fail . happen . help . hesitate .learn . long . mean. manage . offer . ought. plan . prepare. pretend . promise. refuse. seem. tend. wait . wish. undertake.
舉例：
The driver failed to see the other car in time.
司機沒能及時看見另一輛車。
I happen to know the answer to your question.
我碰巧知道你那道問題的答案。

2）動詞+不定式 ；動詞+賓語+不定式
ask, beg, choose, expect , hate, help intend like, love, need prefer, prepare, promise, want, wish…
I like to keep everything tidy. 我喜歡每件東西都保持整潔。
I like you to keep everything tidy. 我喜歡你年使每件東西都保持整潔。
I want to speak to Tom. 我想和湯姆談話。
I want you to speak to Tom. 我想讓你和湯姆談話。

3） 動詞+疑問詞+ to
decide, know, consider forget, learn, remember, show, understand, see, wonder, hear, find out, explain, tell
Please show us how to do that. 請演示給我們如何去做。
There are so many kinds of tape-recorders on sale that I can't make up my mind which to buy.有這么多的錄音機，我都拿不定主意買哪一種。
注意:疑問詞帶不定式在句中作成分時，謂語動詞用單數。
The question is how to put it into practice.
問題是怎樣把它付諸實施。

4)以下動詞後，只能跟不定式作賓語。
如：agree,ask,aim,arrange,choose,decide,demand,expect,fail ,help,hope,lean,long,manage,offer,plan,prepare,
pretend,promise,refuse,wish 等，這些詞大部分可接th at引導的從句。如：
I decided to ask for my money back.
I decided that I would ask for my money back.
When our visit to the farm was over,we expected to startback on foot.
When our visit to the farm was over, we expected that wewould start back on foot.

5)當復合賓語中的賓語是不定式時，先用形式賓語it代替不定式，把不定式置於補語之後，即：主語＋動 詞＋it＋補語＋to do句式。如：
We think it quite important for us to learn a foreignlanguage well.
He feels it his ty to help the poor.

二、作補語
1) 動詞+賓語+不定式(to do)
advise allow appoint believe cause challenge command compel consider declare drive enable encourage find forbid force guess hire imagine impel ince inform instruct invite judge know like order permit persuade remind report request require select send state suppose tell think train trust understand urge warn
例句：
a. Father will not allow us to play on the street.
父親不讓我們在街上玩耍。
b. We believe him to be guilty.
我們相信他是有罪的。

Find 的特殊用法：
Find 後可用分詞做賓補，或先加形式賓語，再加形容詞，最後加帶to 的動詞不定式。find後也可帶一個從句。此類動詞還有get，have。
I found him lying on the ground.
I found it important to learn.
I found that to learn English is important.

2) to + be 的不定式結構，作補語的動詞。
Acknowledge, believe, consider, think, declare(聲稱), discover, fancy(設想), feel find, guess, judge, imagine, know, prove, see(理解), show, suppose, take(以為), understand
We consider Tom to be one of the best students in our class.
我們認為湯姆是班上最好的學生之一。

3) to be +形容詞
Seem, appear, be said, be supposed, be believed, be thought, be known, be reported, hope, wish, desire, want, plan, expect, mean…
The book is believed to be uninteresting.
人們認為這本書沒什麼意思。

4) there be+不定式
believe, expect, intend, like, love, mean, prefer, want, wish, undrstand
We didn't expect there to be so many people there.我們沒料到會有那麼多人在哪裡。
有些動詞需用as 短語做補語，如regard, think believe, take, consider.
We regard Tom as our best teacher. 我們認為湯姆是我們最好的老師。
Mary took him as her father . 瑪麗把他當作自己的父親。

三、作主語
動詞不定式作主語時，句子的謂語動詞常用單數，其位置有以下兩種：
(1)把不定式置於句首。如：
To get there by bike will take us half an hour.

(2)用it作形式主語，把真正的主語不定式置於句後，常用於下列句式中。如：
①It＋be＋名詞＋to do
It's our ty to take good care of the old.

②It takes sb+some time+to do
How long did it take you to finish the work?

③It＋be＋形容詞＋for sb＋to do
It is difficult for us to finish writing the compositionin a quarter of an hour.

④It＋be＋形容詞＋of sb＋to do
It is stupid of you to write down everything the teachersays.

⑤It seems(appears)＋形容詞＋to do
It seemed impossible to save money.

在句型③中，常用表示客觀情況的形容詞，如：difficult,easy,hard,important,impossible,necessary 等；在句型④中，常用careless,clever,good,foolish,honest,kind,lazy,nice,right,silly,stupid,wise等 表示贊揚或批評的詞。在不定式前的sb，可看作其邏輯主語。這一句式有時相當於Sb is＋形容詞＋to do句式 ，如：It'skind of you to help me with my English.=You are kind to helpme with my English.

(3)舉例
1） It's easy (for me) to do that.我做這事太容易了
easy, difficult, hard, important, possible, impossible, comfortable, necessary, better;
the first, the next, the last, the best, too much, too little, not enough
It's so nice to hear your voice.
聽到你的聲音真高興。
It's necessary for you to lock the car when you do not use it.
當你不用車的時候，鎖車是有必要的。

2) It's very kind of you to help us. 他幫助我們，他真好。
Kind, nice, stupid, rude, clever, foolish, thoughtful, thoughtless, brave, considerate(考慮周到的), silly, selfish(自私的)
例句:
It was silly of us to believe him. 我們真愚蠢，竟然相信了他。
It seemed selfish of him not to give them anything. 他不給他們任何東西，這顯得太自私了。

注意：1) 其他系動詞如，look，appear等也可用於此句型
2) 不定式作為句子成分時，動詞用單數形式。
3) 當不定式作主語的句子中又有一個不定式作表語時，不能用It is… to…的句型
(對)To see is to believe. 百聞不如一見。
(錯)It is to believe to see.

四、作表語
不定式作表語表示具體動作或將來動作；動名詞作表語表示抽象的一般行為。
①To be kind to the enemy is to be cruel to the people.
②My chief purpose is to point out the difficulties ofthe matter.
③What I would suggest is to put off the meeting.
當主語和表語都是不定式時，其含義一是條件，一是結果（例①）。當主語是aim,ty,hope,idea,mista ke,plan,purpose,suggestion等為中心詞的名詞片語（例②）時，或以what引導的名詞性分句（例③），不定 式說明主語的內容。
④Our work is serving the people.
⑤What he likes is taking a walk after supper.
⑥The story told by Mr.Wang is interesting.
④⑤句動名詞作表語，與主語部分可以轉換，如Serving thepeople is out work,而⑥句中是現在分詞作 表語，說明主語的性質、狀態，現在分詞具有形容詞的各種特徵，另外，動名詞作表語還應與進行時態區別開 來。

五、作定語
⒈不定式作定語
不定式在句中作定語，置於被修飾的名詞或代詞之後。如：
①The next train to arrive is from Washington.
②Have you anything to be taken to your sister?
③Do you have anything to say on the question?
④Would you please give me some paper to write on?
⑤My wish to visit France has come true at last.

不定式短語作定語和被修飾詞之間表示以下關系：
(1)表示將來的動作（例①）。
(2)與被修飾詞之間有動賓關系，如是不及物動詞，則需加介詞（例④）。
(3)與被修飾詞之間有動賓關系，同時與句中其它詞之間又有邏輯上的主謂關系時，盡管有被動含義，卻仍 用主動語態（例③）；如只有動賓關系，而無邏輯上的主謂關系，則需用被動語態（例②）。
(4)不定式作定語時，一般可轉換為定語從句，例①to arrive=that will arrive。

六、作狀語
1）目的狀語
To… only to (僅僅為了), in order to, so as to, so(such)… as to… (如此……以便……)
He ran so fast as to catch the first bus. 他飛快地跑以便趕上第一班車。
I come here only to say good-bye to you. 我來僅僅是向你告別。

2）作結果狀語，表事先沒有預料到的，要放在句子後面。
What have I said to make you angry.
He searched the room only to find nothing.

3) 表原因
I'm glad to see you.

省to 的動詞不定式
1） 情態動詞 ( 除ought 外，ought to)：
2） 使役動詞 let, have, make：
3） 感官動詞 see, watch, look at, notice , observe, hear, listen to, smell, feel, find 等後作賓補，省略to。

注意：在被動語態中則to 不能省掉。
I saw him dance.
=He was seen to dance.
The boss made them work the whole night.
=They were made to work the whole night.
4） would rather，had better：
5） Why… / why not…：
6） help 可帶to，也可不帶to, help sb (to) do sth：
7） but和except：but前是動詞do時，後面出現的動詞用不帶to的動詞不定式。
8） 由and, or和than連接的兩個不定式，第二個to 可以省去：
9） 通常在discover, imagine, suppose, think, understand等詞後，可以省去to be：
He is supposed (to be) nice. 他應該是個好人。
舉例：
He wants to move to France and marry the girl.
He wants to do nothing but go out.
比較： He wants to do nothing but go out.
He wants to believe anything but to take the medicine.

動詞不定式的否定式
Tell him not to shut the window…
She pretended not to see me when I passed by. 我走過的時候，她假裝沒看見。

不定式的特殊句型
1、不定式的特殊句型so as to
1） 表示目的；它的否定式是so as not to do。
Tom kept quiet about the accident so as not to lose his job.
湯姆對事故保持沉默是為了不丟掉他的工作。
Go in quietly so as not to wake the baby.
輕點進去，別驚醒了嬰兒。
2) so kind as to ---勞駕
Would you be so kind as to tell me the time?
勞駕，現在幾點了。
2、 "Why not +動詞原形"表達向某人提出建議
"為什麼不……?" "干嗎不……?"
例如：
Why not take a holiday?
干嗎不去度假?

3、It's for sb.和 It's of sb.

1）for sb. 常用於表示事物的特徵特點，表示客觀形式的形容詞，如easy, hard, difficult, interesting, impossible等：
It's very hard for him to study two languages. 對他來說學兩門外語是很難的。
2）of sb的句型一般用表示人物的性格，品德，表示主觀感情或態度的形容詞，如good, kind, nice, clever, foolish, right。
It's very nice of you to help me. 你來幫助我，你真是太好了。

for 與of 的辨別方法：
用介詞後面的代詞作主語，用介詞前邊的形容詞作表語，造個句子。如果道理上通順用of，不通則用for。如：
You are nice. (通順，所以應用of)。
He is hard. (人是困難的，不通，因此應用for。)

4、不定式的特殊句型too…to…

1）too…to 太…以至於…
He is too excited to speak.
他太激動了，說不出話來。
---- Can I help you ? 需要我幫忙嗎?
---- Well, I'm afraid the box is too heavy for you to carry it, but thank you all the same. 不用了。這箱子太重，恐怕你搬不動。謝謝您。
2) 如在too前有否定詞，則整個句子用否定詞表達肯定， too 後那個詞表達一種委婉含義，意 為"不太"。
It's never too late to mend. (諺語)
改過不嫌晚。
3) 當too 前面有only, all, but時，意思是：非常… 等於very。
I'm only too pleased to be able to help you. 我非常高興能幫助你。
He was but too eager to get home. 他非常想回家。

動詞不定式的"to"與介詞的"to"區別to 有兩種用法：
一為不定式+動詞原形；
一為介詞+名詞/動名詞, to 在下面的用法中是第二種，即to+ 名詞/動名詞：
admit to承認，confess to承認，be accustomed to 習慣於，be used to 習慣於， stick to 堅持， turn to開始，著手於， devote oneself to 獻身於， be devoted to 致力於， look forward to 盼望，pay attention to 注意

介詞but,except,besides+to do(do)
在這種句型中，如介詞前有動詞do，後面應接不帶to的不定式；如無do，則接to不定式，即帶do不帶to， 帶to不帶do。如：
The enemy soldiers had no choice but to give in.
On Sunday afternoon I had nothing to do but watch TV.

動詞不定式與動名詞區別與聯系
1） 動名詞與不定式的區別：
動名詞表達的是： 狀態，性質，心境，抽象，經常性，已發生的 不定式表達的是： 目的，結果，原因，具體，一次性，將發生的

2） 接不定式或動名詞，意義相同。

3)在下列情況下，一般要用不定式：
①hate,like,love前有would(should)時，如：I'd like to have a cup of coffee.
②當謂語動詞begin,continue,start等是進行式時，如：Thestudents are starting to work on the di fficult maths problem.
③begin,continue,start與know,understand等狀態動詞連用時，如：I soon began to understand what was happening.
⒋advise,allow,encourage,forbid,permit等動詞後接動名詞作賓語，或帶不定式作賓語補足語。如：
Our teachers don't permit our swimming in the lake.
Our teachers don't permit us to swim in the lake.
4） 部分動詞後接不定式或動名詞時，意義差別較大，應根據句子語境選擇使用。
①forget,remember,regret後接不定式，表示現在或未來的動作，接動名詞表示動作已經發生。如：
Don't forget fo post the letter for me.
Have you forgotten meeting her in Beijing Airport?
Remember to close the windows before you leave.
I remember writing him a letter a year ago.
We regret to tell you that all of you are not invited toattend the meeting.
They regretted ordering these books from abroad.

動名詞與不定式語義不同的有11 組：
1 stop to do stop doing
2 forget to do forget doing
3 remember to do remember doing
4 regret to do regret doing
5 cease to do cease doing
6 try to do try doing
7 go on to do go on doing
8 afraid to do afraid doing
9 interested to do interested doing
10 mean to do mean doing
11 begin/ start to do begin/ start doing

1 forget doing/to do
forget to do 忘記要去做某事。 (未做)
forget doing 忘記做過某事。 (已做)
The light in the office is stil on. He forgot to turn it off.
辦公室的燈還在亮著，它忘記關了。(沒有做關燈的動作)
He forgot turning the light off.
他忘記他已經關了燈了。 ( 已做過關燈的動作)
Don't forget to come tomorrow.
別忘了明天來。 (to come動作未做)

2 stop doing/to do
stop to do 停止，中斷做某事後去做另一件事。
stop doing 停止做某事。
They stop to smoke a cigarette. 他們停下來，抽了根煙。
I must stop smoking.. 我必須戒煙了。

3 remember doing/to do
remember to do 記得去做某事 (未做)
remember doing 記得做過某事 (已做)
Remember to go to the post office after school.
記著放學後去趟郵局。
Don't you remember seeing the man before?
你不記得以前見過那個人嗎?

4 regret doing/to do
regret to do 對要做的事遺憾。 (未做)
regret doing 對做過的事遺憾、後悔。 (已做)
I regret to have to do this, but I have no choice.
我很遺憾必須這樣去做，我實在沒有辦法。
I don't regret telling her what I thought.
我不為告訴她我的想法而後悔。

5 cease doing/to do
cease to do 長時間，甚至永遠停做某事。
cease doing 短時停止做某事，以後還會接著做。
That department has ceased to exist forever.
那個部門已不復存在。
The girls ceased chatting for a moment when their teacher passed by.
姑娘們在老師走過時，停了會聊天。

6 try doing/to do
try to do 努力，企圖做某事。
try doing 試驗，試著做某事。
You must try to be more careful.
你可要多加小心。

7 go on doing/to do
go on to do 做了一件事後，接著做另一件事。
go on doing 繼續做原來做的事。
After he had finished his maths，he went on to do his physics.
做完數學後，他接著去做物理。
Go on doing the other exercise after you have finished this one.
作完這個練習後，接著做其他的練習

8 be afraid doing/to do
be afraid to do 不敢，膽怯去做某事，是主觀上的原因不去做，意為"怕"；
be afraid of doing 擔心出現doing的狀況、結果。 doing 是客觀上造成的，意為"生怕，恐怕"。
She was afraid to step further in grass because she was afraid of being
bitten by a snake.
她生怕被蛇咬著，而不敢在草叢中再走一步。

.9 be interested doing/to do
interested to do 對做某事感興趣，想了解某事。
interested in doing 對某種想法感興趣，doing 通常為想法。
I shall be interested to know what happens.
我很想知道發生了什麼事。 (想了解)
I'm interested in working in Switzerland. Do you have any idea about that?
我對在瑞士工作感興趣。你想過這事嗎? (一種想法)

10 mean to doing/to do
mean to do 打算、想
mean doing 意味著
I mean to go, but my father would not allow me to.
我想去，但是我父親不肯讓我去。
To raise wage means increasing purchasing power.
贈加工資意味著增加購買力。

11 begin(start) doing/to do
begin / start to do sth
begin / start doing sth.
1) 談及一項長期活動或開始一種習慣時，使用doing.
How old were you when you first started playing the piano?
你幾歲時開始彈鋼琴？
2) begin, start用進行時時，後面動詞用不定式to do
I was beginning to get angry。
我開始生起氣來。
3) 在attempt, intend, begin, start 後接know, understand, realize這類動詞時，常用不定式to
do。
I begin to understand the truth。
我開始明白真相。
4) 物作主語時
It began to melt.

12 感官動詞 + doing/to do
感官動詞 see, watch, observe, notice, look at, hear, listen to, smell, taste,
feel + do 表示動作的完整性，真實性；+doing 表示動作的連續性，進行性
I saw him work in the garden yesterday.
昨天我看見他在花園里幹活了。(強調"我看見了"這個事實)
I saw him working in the garden yesterday.（強調"我見他正幹活"這個動作）
昨天我見他正在花園里幹活。

E. 不等式的運演算法則

「不等式
運演算法則
為不等式兩邊相加或相減同一個數或式子，
不等號
的方向不變。不等式兩邊相乘或相除同一個
正數
，不等號的方向不變。不等式兩邊乘或除以同一個負數，不等號的方向改變。」

F. 洛必達法則其他不定式

摘要 洛必達法則是在一定條件下通過分子分母分別求導再求極限來確定未定式值的方法 [1] 。眾所周知，兩個無窮小之比或兩個無窮大之比的極限可能存在，也可能不存在。因此，求這類極限時往往需要適當的變形，轉化成可利用極限運演算法則或重要極限的形式進行計算。洛必達法則便是應用於這類極限計算的通用方法 [2] 。

G. 不等式3個運演算法則如何正確使用

不等式有三種：
（1）基本不等式 設a>b則

• ac>bc(c>0)；

• ac<bc(c<0)；

• a/c>b/c(c>0)；

• a/c<b/c(c<0)；

• a^n>b^n（a>0，b>0，n>0)

（2）絕對不等式

（3）絕對值不等式

• |A+B|≤|A|+|B|；

• |A-B|≤|A|+|B|；

• |A-B|≥|A|-|B|；

• -|A|≤A≤|A|；

• √(A²)=|A|；

• |AB|=|A||B|；

• |A/B|=|A|/|B|
  
  

H. 洛比達法則適用於0/0、∞/∞型不定式，等價無窮小量不適用於不定式，請問啥是不定式啊

能用極限運演算法則(四則運算，復合合函數求極限法則等)直接計算的可以叫定式，相對的，不能直接通過極限運演算法則計算的就叫不定式,主要有以下形式：

∞-∞,∞*0,0/0,∞/∞,1^∞,0^0,∞^0.

計算上往往把它們化為0/0,∞/∞這兩種類型，以期利用洛必達法則。